The radius of a sphere with a surface area of 616 cm² is 7 cm.
Understanding the dimensions of a sphere, particularly its radius, is fundamental in various scientific and engineering applications, from designing spherical tanks to calculating planetary volumes. The surface area of a sphere, which represents the total area on its outer boundary, is directly related to its radius through a simple mathematical formula.
Understanding Sphere Surface Area
The surface area of a sphere is the measure of the total area that the surface of the sphere occupies. It's akin to the amount of material needed to cover the entire outside of the sphere. This property is crucial in fields like physics, where surface-to-volume ratios impact heat transfer, or in chemistry, for understanding the properties of nanoparticles.
The formula for the surface area ($A$) of a sphere is:
$A = 4 \pi r^2$
Where:
- $A$ is the surface area of the sphere
- $\pi$ (pi) is a mathematical constant approximately equal to 3.14159 or $\frac{22}{7}$
- $r$ is the radius of the sphere
Calculating the Radius
To find the radius of a sphere when its surface area is known, we can rearrange the formula.
Given a surface area of 616 cm², we can solve for $r$:
- Start with the formula:
$A = 4 \pi r^2$ - Substitute the given surface area:
$616 \text{ cm}^2 = 4 \pi r^2$ - Isolate $r^2$:
$r^2 = \frac{616}{4 \pi}$
$r^2 = \frac{154}{\pi}$ - Use the approximation for $\pi$ (e.g., $\frac{22}{7}$ for a precise fit):
$r^2 = \frac{154}{\frac{22}{7}}$
$r^2 = 154 \times \frac{7}{22}$
$r^2 = 7 \times 7$
$r^2 = 49$ - Calculate the radius ($r$):
$r = \sqrt{49}$
$r = 7 \text{ cm}$
This calculation confirms that for a sphere with a surface area of 616 cm², the radius is exactly 7 cm.
Summary of Sphere Dimensions
Here's a quick overview of the sphere's dimensions based on the given surface area:
| Property | Value | Unit |
|---|---|---|
| Surface Area (A) | 616 | cm² |
| Radius (r) | 7 | cm |
Practical Applications of Sphere Calculations
Understanding how to calculate a sphere's radius from its surface area has numerous practical applications across various industries:
- Manufacturing: Determining the amount of coating or paint needed for spherical objects like ball bearings, chemical storage tanks, or decorative globes.
- Packaging: Estimating the material required for spherical packaging.
- Science and Research: Calculating the surface-to-volume ratio of cells, bubbles, or planets for studies in biology, physics, and astronomy.
- Architecture and Design: Planning the materials for spherical domes or artistic installations.
For further exploration of sphere geometry and related formulas, you can refer to resources like Khan Academy's Geometry lessons or Wikipedia's page on Spheres. These resources offer detailed explanations and examples for various geometric shapes.
